All throughout the preface and the first chapter Feser made numerous extremely bold claims that he promises to back up in the later chapters. By chapter two, entitled Greeks Bearing Gifts, we start seeing some of those justifications come to light. The chapter starts out on a crash course through ancient Greek philosophy leading up to Plato and then Aristotle. I won't summarize Feser's teaching unless I think it is significant for his objective, which is to show that "a certain kind of" religion and god are not only reasonable to believe in, but that it's logically impossible that naturalism is true.
Plato and Aristotle are considered to be two of the greatest philosophers of all time, and I would largely agree. That's not to say that I agree with all of their ideas, especially their metaphysical ones, it's just to recognize the fact that they were both deeply analytic thinkers and widely influential. For example, I regard the Euthyphro Dilemma, from Plato's Euthyphro, as one of the greatest pieces of moral insight. But, I digress. For Feser, he focuses first on Plato's Theory of Forms, which is one of the things I think Plato got wrong.
Take the triangle. Any triangle physically drawn or created will in some way be imperfect, if only by a tiny amount. They will all lack features that perfectly exemplify a triangle - that is, they will have features not part of a triangle's essence or nature. Plato argues from this that the essence or nature of triangularity is not material or known through our senses, and when we exemplify triangles physically they go in and out of existence, but its essence stays the same. The essential features of triangularity are therefore according to Plato, universal, and not particular, immaterial, and not material, and known through the intellect and not through the senses.
Feser is making the case for Platonic realism, and makes arguments against nominalism, and conceptualism. Platonic realism is the view that universals (like triangles, squares, and other geometric patterns) and abstract objects (like numbers) exist independently of minds or physical space and time. Nominalism is the view that these objects do not exist, and conceptualism is the view that these objects exist, but only as concepts in our minds. Feser presents several arguments to try and show that realism is true and that nominalism and conceptualism are false. The reason why he's trying to do so starts becoming clear on page 36 where he writes:
A triangle is a triangle only because it participates in the Form of Trianglarity; a squirrel is a squirrel only because it participates in the Form of Squirrel; and so forth. By the same token, something is going to count as a better triangle the more perfectly it participates in or instantiates triangularity, and a squirrel would be a better squirrel the more perfectly it participates in or instantiates the Form of Squirrel.
This is all leading up to the natural law theory of ethics that many Catholics, like Feser, think forms the basis of our morality. Feser goes on: